pre-prints (linked to the arXiv)

• • (with Grace Garden and Ben Martin) Essential tori in 3-manifolds not detected in any characteristic (arXiv), submitted for publication.
• • (with Grace Garden) An invitation to Culler-Shalen theory in arbitrary characteristic (arXiv), submitted for publication.
• • (with Marc Kegel, Arunima Ray, Em Thompson and Jonathan Spreer) On a volume invariant of 3-manifolds (arXiv), submitted for publication.
• • (with Daryl Cooper) The space of strictly-convex real-projective structures on a closed manifold (arXiv), submitted for publication.
• • (with Daryl Cooper) On properly convex real-projective manifolds with generalized cusps (arXiv), submitted for publication.
• • (with Daryl Cooper) The space of properly-convex structures (arXiv), submitted for publication.
• • (with Robert Haraway, Robert Löwe and Dominic Tate) On Moduli Spaces of Convex Projective Structures on Surfaces: Outitude and Cell-Decomposition in Fock-Goncharov Coordinates (arXiv).
• • (with Ben Burton) Computing closed essential surfaces in 3-manifolds (arXiv), submitted for publication.

in press

• • (with Youheng Yao) On the topology of character varieties of once-punctured torus bundles (arXiv), to appear in Algebraic and Geometric Topology.
• • (with Bus Jaco, Hyam Rubinstein and Jonathan Spreer) Complexity of 3-manifolds obtained by Dehn filling (arXiv), to appear in Algebraic and Geometric Topology.
• • (with Bus Jaco, Hyam Rubinstein and Jonathan Spreer) Slope norm and an algorithm to compute the crosscap number (arXiv), to appear in Algebraic and Geometric Topology.
• • (with Daryl Cooper and William Worden) The Thurston norm via spun-normal immersions (arXiv), to appear in Transactions of the American Mathematical Society.

books

1. (with Olivier Collin, Stefan Friedl, Cameron Gordon and Liam Watson, eds.) Characters in low-dimensional topology (publisher), Contemporary Mathematics, 760, Centre de Recherches Mathématiques Proceedings. American Mathematical Society, Providence, RI, 2020.
2. (with Alex Casella and Dominic Tate) Moduli spaces of real projective structures on surfaces (publisher), MSJ Memoirs, 38. Mathematical Society of Japan, Tokyo, 2020. A shorter version is available at the arXiv.
3. (with Craig Hodgson, Bus Jaco and Marty Scharlemann, eds.) Geometry and Topology Down Under (publisher), Contemp. Math., 597, Amer. Math. Soc., Providence, RI, 2013.

refereed publications in journals or conference proceedings

1. (with Hyam Rubinstein and Jonathan Spreer) A new family of minimal ideal triangulations of cusped hyperbolic 3-manifolds (publisher, arXiv). In: Wood, D.R., de Gier, J., Praeger, C.E. (eds) 2021-2022 MATRIX Annals, 5028, MATRIX Book Series, vol 5. Springer, Cham., 2024.
2. (with Jonathan Spreer) Determining the trisection genus of orientable and non-orientable PL 4-manifolds through triangulations (publisher, arXiv), Experimental Mathematics 31 (2022), no. 3, 897-907. Expanded version of a conference paper.
3. (with Joan Porti) Projective structures on a hyperbolic 3-orbifold (arXiv), Acta Mathematica Vietnamica 46 (2021), no. 2, 347-355. (Proceedings of Quantum Topology and Hyperbolic Geometry Conference).
4. Tropical varieties associated to ideal triangulations: The Whitehead link complement (publisher, arXiv), 293-339, Contemp. Math., 760, Amer. Math. Soc., Providence, RI, 2020
5. (with Hyam Rubinstein) Multisections of piecewise linear manifolds (publisher, arXiv), Indiana University Mathematics Journal 69 (2020), 2209-2239
6. (with Stefan Friedl) Two-generator one-relator groups and marked polytopes (publisher, arXiv), Annales de l'Institut Fourier, (Grenoble) 70 (2020), no. 2, 831-880.
7. (with Bus Jaco and Hyam Rubinstein and Jonathan Spreer) Z2-Thurston Norm and Complexity of 3-Manifolds, II (publisher, arXiv), Algebraic & Geometric Topology 20-1 (2020), 503-529.
8. (with Alex Casella and Charlie Katerba) Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds (publisher, arXiv), Proceedings of the American Mathematical Society 148 (2020), no. 5, 2257-2271.
9. (with Bus Jaco and Hyam Rubinstein and Jonathan Spreer) On minimal ideal triangulations of cusped hyperbolic 3-manifolds (arXiv, publisher), Journal of Topology 13 (2020), no. 1, 308-342.
10. (with Hyam Rubinstein and Henry Segerman) Traversing three-manifold triangulations and spines (publisher, arXiv), L'Enseignement Mathématique (2) 65 (2019), 155-206.
11. (with Michelle Chu) Reflections on trisection genus (arXiv, publisher), Revue Roumaine de Mathématiques Pures et Appliquées 64 (2019), 4, 395-402 (Proceedings of JARCS 2017).
12. (with Stefan Friedl and Wolfgang Lück) Groups and polytopes (arXiv), Breadth in contemporary topology, 57-77, Proc. Sympos. Pure Math., 102, Amer. Math. Soc., Providence, RI, 2019.
13. (with Robert C. Haraway III) Tessellating the moduli space of strictly convex projective structures on the once-punctured torus (publisher, arXiv), Experimental Mathematics 28 (2019), no. 3, 369-384.
14. (with Mark Bell, Joel Hass, and Hyam Rubinstein) Computing trisections of 4-manifolds (publisher, arXiv), Proceedings of the National Academy of Sciences of the United States of America, October 23, 2018, 115 (43) 10901-10907.
15. (with Hyam Rubinstein) Generalised trisections in all dimensions (publisher, preprint), Proceedings of the National Academy of Sciences of the United States of America, October 23, 2018, 115 (43) 10908-10913.
16. (with Stefan Friedl and Montek Gill) Linear representations of 3-manifold groups over rings (publisher, arXiv), Proceedings of the American Mathematical Society 146 (2018), no. 11, 4951-4966.
17. (with Jonathan Spreer) The trisection genus of standard simply connected PL 4-manifolds (publisher, arXiv), 34nd International Symposium on Computational Geometry (SoCG 2018). In Leibniz International Proceedings in Informatics (LIPICS), vol. 99, pg. 71:1-71:13, 2018.
18. (with Jonathan Spreer), Unravelling the Dodecahedral Spaces (publisher, arXiv). In: Wood D., de Gier J., Praeger C., Tao T. (eds) 2016 MATRIX Annals, 323–347, MATRIX Book Series, vol 1., Springer, Cham, 2018.
19. (with Daryl Cooper and Darren Long) Deforming convex projective manifolds (publisher, arXiv), Geometry & Topology 22 (2018) 1349-1404.
20. (with Hsien-Chih Chang, Jeff Erickson, David Letscher, Arnaud de Mesmay, Saul Schleimer, Eric Sedgwick, Dylan Thurston) Tightening Curves on Surfaces via Local Moves (publisher), SODA 2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, 2018, 121-135.
21. (with Stefan Friedl and Kevin Schreve) Thurston norm via Fox calculus (arXiv), Geometry & Topology 21 (2017), no. 6, 3759-3784.
22. (with Feng Luo) A new combinatorial class of 3-manifold triangulations (arXiv, publisher), Asian Journal of Mathematics 21 (2017), no.3, 543-570.
23. (with Alex Casella and Feng Luo) Pseudo-developing maps for ideal triangulations II: Positively oriented ideal triangulations of cone-manifolds (arXiv), Proceedings of the American Mathematical Society 145 (2017), 3543-3560.
24. (with Ben Klaff) A birationality result for character varieties (arXiv, publisher), Mathematical Research Letters 23 (2016), no.4, 1099-1110.
25. (with Bus Jaco, Jesse Johnson and Jonathan Spreer), Bounds for the genus of a normal surface (arXiv, publisher), Geometry & Topology 20 (2016) 1625–1671.
26. (with Sampson Wong) An algorithm for the Euclidean cell decomposition of a cusped strictly convex projective surface (arXiv, publisher), Journal of Computational Geometry 7 (2016), no.1, 237–255.
27. (with Ben Burton, Murray Elder and Arkadius Kalka) 2-manifold recognition is in logspace (arXiv, publisher), Journal of Computational Geometry 7 (2016), no.1, 70-85.
28. (with Craig Hodgson, Hyam Rubinstein and Henry Segerman) Triangulations of 3-manifolds with essential edges (arXiv, publisher), Annales de la Faculté des Sciences de Toulouse 24 (2015), no.5, 1103-1145 (special issue dedicated to Michel Boileau).
29. (with Daryl Cooper) Complexity functions on 1-dimensional cohomology (arXiv, publisher), Revue Roumaine de Mathématiques Pures et Appliquées 60 (2015), no. 4, 535-550 (Proceedings of JARCS 2013).
30. (with Hyam Rubinstein) Even triangulations of n-dimensional pseudo-manifolds (arXiv, publisher), Algebraic and Geometric Topology 15 (2015), no. 5, 2949–2984.
31. (with Daryl Cooper and Darren Long) On Convex Projective Manifolds and Cusps (arXiv), Advances in Mathematics 277 (2015), 181-251.
32. (with Ben Burton and Alexander Coward) Computing closed essential surfaces in knot complements (arXiv, publisher), SoCG '13: Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry, ACM, 2013, 405-414.
33. (with Bus Jaco and Hyam Rubinstein) Z2-Thurston Norm and Complexity of 3-Manifolds (arXiv), Mathematische Annalen 356 (2013) no. 1, 1-22.
34. (with Feng Luo and Tian Yang) Thurston's Spinning Construction and Solutions to the Hyperbolic Gluing Equations for Closed Hyperbolic 3-Manifolds (arXiv), Proceedings of the American Mathematical Society 141 (2013), no. 1, 335-350.
35. Degenerations of ideal hyperbolic triangulations (arXiv), Mathematische Zeitschrift 272 (2012), no. 3-4, 793-823.
36. (with Ben Burton and Hyam Rubinstein) The Weber-Seifert dodecahedral space is non-Haken (arXiv, publisher), Transactions of the American Mathematical Society 364 (2012), no. 2, 911-932.
37. (with Craig Hodgson, Hyam Rubinstein and Henry Segerman) Veering triangulations admit strict angle structures (arXiv), Geometry & Topology 15 (2011) 2073-2089.
38. (with Henry Segerman) Pseudo-developing maps for ideal triangulations I: Essential edges and generalised hyperbolic gluing equations (arXiv), in Topology and geometry in dimension three, 85-102, Contemp. Math., 560, Amer. Math. Soc., Providence, RI, 2011.
39. (with Bus Jaco and Hyam Rubinstein) Coverings and Minimal Triangulations of 3-Manifolds (arXiv), Algebraic and Geometric Topology 11 (2011) 1257-1265.
40. (with Bus Jaco and Hyam Rubinstein) Minimal triangulations for an infinite family of lens spaces (arXiv), Journal of Topology 2 (2009), no. 1, 157-180.
41. (with Daryl Cooper) Thurston norm via normal surfaces (arXiv), Pacific Journal of Mathematics 239 (2009) 1-15.
42. Normal surfaces in topologically finite 3-manifolds (arXiv), L'Enseignement Mathématique, 54 (2008) 329-380.
43. (with Feng Luo and Saul Schleimer) Geodesic ideal triangulations exist virtually (arXiv), Proceedings of the American Mathematical Society 136 (2008) 2625-2630.
44. (with Feng Luo) Angle structures and normal surfaces (arXiv), Transactions of the American Mathematical Society 360 (2008) 2849-2866.
45. (with Eric Chesebro) Not all boundary slopes are strongly detected by the character variety (arXiv), Communications in Analysis and Geometry 15 (2007) 695-723.
46. Boundary slopes and the logarithmic limit set (arXiv) the paper (minus the example) appeared in Topology 44 (2005) 203-216.
47. Character varieties of mutative 3-manifolds (arXiv), Algebraic and Geometric Topology 4 (2004) 133-149.
48. On the Kinoshita-Terasaka knot and generalised Conway mutation (223 KB), Journal of Knot Theory and its Ramifications 9 (2000) 557-575.

non-refereed contributions to conference proceedings

• • Structure of 0–efficient or minimal triangulations (publisher), Oberwolfach Reports 9 (2012), no. 2, 1474-1478.
• • Applications of tropical geometry to groups and manifolds (publisher), Oberwolfach Reports 4 (2007), no. 4, 3281-3283.

extended abstracts in conference proceedings

• • (with Ben Burton, Arne Hillebrand, Maarten Löffler, Saul Schleimer, Dylan Thurston and Wouter van Toll) On the Topology of Walkable Environments, paper #66 in EuroCG 2018 Proceedings of the 34th European Workshop on Computational Geometry (EuroCG).

survey or expository papers

• • (with Jonathan Bowden, Diarmuid Crowley, Jim Davis, Stefan Friedl, Carmen Rovi) Open problems in the topology of manifolds. In: Wood D., de Gier J., Praeger C., Tao T. (eds) 2019-20 MATRIX Annals, 647–659, MATRIX Book Ser., 4, Springer, Cham., 2021.
• • Warum glücklich statt einfach nur normal? (publisher), Mitteilungen der DMV 24 (2016), no. 4, 197–207.

conference reports

• • (with Benjamin Burton, Herbert Edelsbrunner and Jeff Erickson) Computational Geometric and Algebraic Topology. Abstracts from the workshop held October 11–17, 2015. Oberwolfach Reports 12 (2015), no. 4, 2637-2699.
• • Computational & Algorithmic Topology in Sydney, Gazette of the Australian Mathematical Society, Volume 41, Number 4 (2014) 229-230.
• • Geometry & Topology Down Under, Gazette of the Australian Mathematical Society, Volume 38, Number 5 (2011) 261-263.

course notes

• Finding hyperbolic structures of and surfaces in 3-manifolds (pdf file, 747 KB, 82 pages), notes on Culler-Shalen theory, 2003.

thesis

• On character varieties: Surfaces associated to mutation & deformation of hyperbolic 3-manifolds, PhD thesis (supervised by Craig Hodgson and Walter Neumann), The University of Melbourne (2002).
  The material has been expanded and published in the following research articles:

  • On the Kinoshita-Terasaka knot and generalised Conway mutation (223 KB), Journal of Knot Theory and its Ramifications 9 (2000) 557-575.

  • Character varieties of mutative 3-manifolds (arXiv), Algebraic and Geometric Topology 4 (2004) 133-149.

  • Boundary slopes and the logarithmic limit set (arXiv) the paper (minus the example) appeared in Topology 44 (2005) 203-216.

  • Degenerations of ideal hyperbolic triangulations (arXiv), Mathematische Zeitschrift 272 (2012), no. 3-4, 793-823.

  • Tropical varieties associated to ideal triangulations: The Whitehead link complement (arXiv), to appear in Characters in low-dimensional topology, Contemporary Mathematics, Centre de Recherches Mathématiques Proceedings.

reviews (linked to zbMath)

• Zbl 1378.57026 Neil R. Hoffman and Jessica S. Purcell: Geometry of planar surfaces and exceptional fillings. Bull. Lond. Math. Soc. 49, No. 2, 185-201 (2017).
• Zbl 1375.57011 Joshua A. Howie: A characterisation of alternating knot exteriors. Geom. Topol. 21, No. 4, 2353-2371 (2017).
• Zbl 06633013 Daryl Cooper and William Goldman: A 3-manifold with no real projective structure. Ann. Fac. Sci. Toulouse, Math. (6) 24, No. 5, 1219-1238 (2015).

reviews (linked to MathSciNet; subscription only)

• MR2525782 David Gabai, Robert Meyerhoff and Peter Milley: Minimum volume cusped hyperbolic three-manifolds. J. Amer. Math. Soc. 22 (2009), no. 4, 1157--1215.
• MR2423275 Bernard Perron: Filtration de Johnson et groupe de Torelli modulo $p$, $p$ premier. (French). C. R. Math. Acad. Sci. Paris 346 (2008), no. 11-12, 667--670.
• MR2390346 Steven Boyer, Marc Culler, Peter Shalen and Xingru Zhang: Characteristic subsurfaces, character varieties and Dehn fillings. Geom. Topol. 12 (2008), no. 1, 233--297.
• MR2349300 Hitoshi Murakami: The colored Jones polynomials and the Alexander polynomial of the figure-eight knot. JP J. Geom. Topol. 7 (2007), no. 2, 249--269.
• MR2329768 Kazuhiro Hikami: Asymptotics of the colored Jones polynomial and the A-polynomial. Nuclear Phys. B 773 (2007), no. 3, 184--202.
• MR2323541 Hitoshi Murakami: A version of the volume conjecture. Adv. Math. 211 (2007), no. 2, 678--683.
• MR2263219 Ian Agol, Marc Culler and Peter Shalen: Dehn surgery, homology and hyperbolic volume. Algebr. Geom. Topol. 6 (2006), 2297--2312.
• MR2263219 Stefan Friedl and Taehee Kim: The Thurston norm, fibered manifolds and twisted Alexander polynomials. Topology 45 (2006), no. 6, 929--953.
• MR2243744 Jean-Baptiste Meilhan: Goussarov-Habiro theory for string links and the Milnor-Johnson correspondence. Topology Appl. 153 (2006), no. 14, 2709--2729.
• MR2191691 Ser Peow Tan, Yan Loi Wong and Ying Zhang: The SL(2,C) character variety of a one-holed torus. Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 103--110 (electronic).
• MR2199344 Dmitriy Derevnin, Alexander Mednykh and Michele Mulazzani: Volumes for twist link cone-manifolds. Bol. Soc. Mat. Mexicana (3) 10 (2004), Special Issue, 129--145.
• MR2187606 Sam Nelson: Virtual crossing realization. J. Knot Theory Ramifications 14 (2005), no. 7, 931--951.
• MR2153977 Shelly Harvey: Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm. Topology 44 (2005), no. 5, 895--945.
• MR2161256 Xingru Zhang: Trace fields of representations and virtually Haken 3-manifolds. Q. J. Math. 56 (2005), no. 3, 431--442.
• MR2135546 Emmanuel Auclair and Christine Lescop: Clover calculus for homology 3-spheres via basic algebraic topology. Algebr. Geom. Topol. 5 (2005), 71--106 (electronic).
• MR2147736 Atsushi Ishii and Taizo Kanenobu: Different links with the same Links-Gould invariant. Osaka J. Math. 42 (2005), no. 2, 273--290.
• MR2116925 Nafaa Chbili: Symétries en dimension trois: une approche quantique. Cubo 6 (2004), no. 4, 133--165.